Related papers: Weak formulation for singular diffusion equation w…
We investigate an initial-(periodic-)boundary value problem for a continuum equation, which is a model for motion of grain boundaries based on the underlying microscopic mechanisms of line defects (disconnections) and integrated the effects…
We establish the global existence of weak solutions for a two-species cross-diffusion system, set on the 1-dimensional flat torus, in which the evolution of each species is governed by two mechanisms. The first of these is a diffusion which…
We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy minimisation approach. More…
In this research note we provide a variational basis for the optimal artificial diffusion method, which has been a cornerstone in developing many stabilized methods. The optimal artificial diffusion method produces exact nodal solutions…
The global existence of bounded weak solutions to a diffusion system modeling biofilm growth is proven. The equations consist of a reaction-diffusion equation for the substrate concentration and a fourth-order Cahn-Hilliard-type equation…
We establish a weak-strong uniqueness principle for the two-phase Mullins-Sekerka equation in the plane: As long as a classical solution to the evolution problem exists, any weak De Giorgi type varifold solution (see for this notion the…
The continuous dependence on the initial data and consequently the uniqueness of bounded weak solutions to a class of triangular reaction-cross-diffusion equations is shown. The class includes two-species doubly degenerate equations for…
We provide a detailed (and fully rigorous) derivation of several fundamental properties of bounded weak solutions to initial-value problems for general conservative 2nd-order parabolic equations with p-Laplacian diffusion and (arbitrary)…
This article is devoted to questions concerning the existence of solutions for partial differential equation problems modeling granular flows. The models studied take into account the complex threshold rheology of these flows, as well as…
A global existence theorem on weak solutions is shown for the continuous coagulation equation with collisional breakage under certain classes of unbounded collision kernels and distribution functions. This model describes the dynamics of…
We consider a class of Cahn-Hilliard equation with kinetic rate dependent dynamic boundary conditions that describe possible short-range interactions between the binary mixture and the solid boundary. In the presence of surface diffusion on…
In this paper, we develop an analytical framework for the partial differential equation underlying the consensus-based optimization model. The main challenge arises from the nonlinear, nonlocal nature of the consensus point, coupled with a…
This paper is concerned with the initial-boundary value problem for an evolutionary variational inequality complying with three intrinsic properties: complete irreversibility, unilateral equilibrium of an energy and an energy conservation…
We investigate uniqueness of weak solutions for a system of partial differential equations capturing behavior of magnetoelastic materials. This system couples the Navier-Stokes equations with evolutionary equations for the deformation…
Thermodynamically consistent models for two-phase flow in porous media have attracted significant attention in recent years. In this paper, we prove the existence, uniqueness and regularity of the weak solution to such a recent model…
In this paper, we consider a class of initial-boundary value problems governed by pseudo-parabolic total variation flows. The principal characteristic of our problem lies in the velocity term of the diffusion flux, a feature that can bring…
In this paper, we are concerned with the stochastic time-fractional diffusion-wave equations in a Hilbert space. The main objective of this paper is to establish properties of the stochastic weak solutions of the initial-boundary value…
The weak-strong uniqueness for Maxwell--Stefan systems and some generalized systems is proved. The corresponding parabolic cross-diffusion equations are considered in a bounded domain with no-flux boundary conditions. The key points of the…
Using an approach by contradiction we prove the existence and uniqueness of a weak solution to a quasi-linear elliptic boundary value problem with singular convection term and Hardy Potential. Whose simplest model is \begin{equation*}…
We introduce a new wave formulation for the relativistic Euler equations with vacuum boundary conditions that consists of a system of non-linear wave equations in divergence form with a combination of acoustic and Dirichlet boundary…